blitz_4.html

A C++ class library for scientific computing

<P>

<DT><CODE>backward32(A,dimension)</CODE>
<DD>3rd derivative, 2nd order accurate.  Factor: <EM>2h^3</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-4</td><td>-3</td><td>-2</td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">3</font></td><td bgcolor="#000000"><font color="#ffffff">-14</font></td><td bgcolor="#000000"><font color="#ffffff">24</font></td><td bgcolor="#000000"><font color="#ffffff">-18</font></td><td bgcolor="#000060"><font color="#ffffff">5</font></td></tr>
</table>
<P>

<DT><CODE>backward42(A,dimension)</CODE>
<DD>4th derivative, 2nd order accurate.  Factor: <EM>h^4</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-5</td><td>-4</td><td>-3</td><td>-2</td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-2</font></td><td bgcolor="#000000"><font color="#ffffff">11</font></td><td bgcolor="#000000"><font color="#ffffff">-24</font></td><td bgcolor="#000000"><font color="#ffffff">26</font></td><td bgcolor="#000000"><font color="#ffffff">-14</font></td><td bgcolor="#000060"><font color="#ffffff">3</font></td></tr>
</table>
</DL>
<P>

Note that the above are available in normalized versions <CODE>backward11n</CODE>,
<CODE>backward21n</CODE>, ..., <CODE>backward42n</CODE> which have factors of <EM>h</EM>,
<EM>h^2</EM>, <EM>h^3</EM>, or <EM>h^4</EM> as appropriate.  
</P><P>

These are available in multicomponent versions: for example,
<CODE>backward42(A,component,dimension)</CODE> gives the backward42 operator for
the specified component (Components are numbered 0, 1, ... N-1).
</P><P>

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<H3> 4.4.4 Laplacian (<EM>nabla ^2</EM>) operators </H3>
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<P>

<DL COMPACT>
<DT><CODE>Laplacian2D(A)</CODE>
<DD>2nd order accurate, 2-dimensional laplacian.  Factor: <EM>h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-1</td><td>0</td><td>1</td></tr><tr align=right><td>-1</td><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td></td></tr>
<tr align=right><td>0</td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td bgcolor="#000060"><font color="#ffffff">-4</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
<tr align=right><td>1</td><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td></td></tr>
</table>
<P>

<DT><CODE>Laplacian3D(A)</CODE>
<DD>2nd order accurate, 3-dimensional laplacian.  Factor: <EM>h^2</EM>
<P>

<DT><CODE>Laplacian2D4(A)</CODE>
<DD>4th order accurate, 2-dimensional laplacian.  Factor: <EM>12h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td>-2</td><td></td><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td></td><td></td></tr>
<tr align=right><td>-1</td><td></td><td></td><td bgcolor="#000000"><font color="#ffffff">16</font></td><td></td><td></td></tr>
<tr align=right><td>0</td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td bgcolor="#000000"><font color="#ffffff">16</font></td><td bgcolor="#000060"><font color="#ffffff">-60</font></td><td bgcolor="#000000"><font color="#ffffff">16</font></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td></tr>
<tr align=right><td>1</td><td></td><td></td><td bgcolor="#000000"><font color="#ffffff">16</font></td><td></td><td></td></tr>
<tr align=right><td>2</td><td></td><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td></td><td></td></tr>
</table>
<P>

<DT><CODE>Laplacian3D4(A)</CODE>
<DD>4th order accurate, 3-dimensional laplacian.  Factor: <EM>12h^2</EM>
</DL>
<P>

Note that the above are available in normalized versions
<CODE>Laplacian2D4n</CODE>, <CODE>Laplacian3D4n</CODE> which have factors <EM>h^2</EM>.
</P><P>

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<H3> 4.4.5 Gradient (<EM>nabla</EM>) operators </H3>
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<P>

These return <CODE>TinyVector</CODE>s of the appropriate numeric type and length:
</P><P>

<DL COMPACT>

<DT><CODE>grad2D(A)</CODE>
<DD>2nd order, 2-dimensional gradient (vector of first derivatives), generated
using the central12 operator.  Factor: <EM>2h</EM>
<P>

<DT><CODE>grad2D4(A)</CODE>
<DD>4th order, 2-dimensional gradient, using central14 operator.  Factor: <EM>12h</EM>
<P>

<DT><CODE>grad3D(A)</CODE>
<DD>2nd order, 3-dimensional gradient, using central12 operator.  Factor: <EM>2h</EM>
<P>

<DT><CODE>grad3D4(A)</CODE>
<DD>4th order, 3-dimensional gradient, using central14 operator.  Factor: <EM>12h</EM>
</DL>
<P>

These are available in normalized versions <CODE>grad2Dn</CODE>, <CODE>grad2D4n</CODE>,
<CODE>grad3Dn</CODE> and <CODE>grad3D4n</CODE> which have factors <EM>h</EM>.
</P><P>

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<H3> 4.4.6 Jacobian operators </H3>
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<P>

The Jacobian operators are defined over 3D vector fields only (e.g.
<CODE>Array&#60;TinyVector&#60;double,3&#62;,3&#62;</CODE>).  They return a
<CODE>TinyMatrix&#60;T,3,3&#62;</CODE> where T is the numeric type of the vector field.
</P><P>

<DL COMPACT>
<DT><CODE>Jacobian3D(A)</CODE>
<DD>2nd order, 3-dimensional Jacobian using the central12 operator.  Factor:
<EM>2h</EM>.
<P>

<DT><CODE>Jacobian3D4(A)</CODE>
<DD>4th order, 3-dimensional Jacobian using the central14 operator.  Factor:
<EM>12h</EM>.
</DL>
<P>

These are also available in normalized versions <CODE>Jacobian3Dn</CODE> and
<CODE>Jacobain3D4n</CODE> which have factors <EM>h</EM>.
</P><P>

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<H3> 4.4.7 Grad-squared operators </H3>
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<P>

There are also grad-squared operators, which return <CODE>TinyVector</CODE>s of
second derivatives:
</P><P>

<DL COMPACT>
<DT><CODE>gradSqr2D(A)</CODE>
<DD>2nd order, 2-dimensional grad-squared (vector of second derivatives),
generated using the central22 operator.  Factor: <EM>h^2</EM>
<P>

<DT><CODE>gradSqr2D4(A)</CODE>
<DD>4th order, 2-dimensional grad-squared, using central24 operator.  Factor:
<EM>12h^2</EM>
<P>

<DT><CODE>gradSqr3D(A)</CODE>
<DD>2nd order, 3-dimensional grad-squared, using the central22 operator.
Factor: <EM>h^2</EM>
<P>

<DT><CODE>gradSqr3D4(A)</CODE>
<DD>4th order, 3-dimensional grad-squared, using central24 operator.  Factor:
<EM>12h^2</EM>
</DL>
<P>

Note that the above are available in normalized versions <CODE>gradSqr2Dn</CODE>,
<CODE>gradSqr2D4n</CODE>, <CODE>gradSqr3Dn</CODE>, <CODE>gradSqr3D4n</CODE> which have factors
<EM>h^2</EM>.
</P><P>

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<H3> 4.4.8 Curl (<EM>nabla times</EM>) operators </H3>
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<P>

These curl operators return scalar values:
</P><P>

<DL COMPACT>
<DT><CODE>curl(Vx,Vy)</CODE>
<DD>2nd order curl operator using the central12 operator.  Factor: <EM>2h</EM>
<P>

<DT><CODE>curl4(Vx,Vy)</CODE>
<DD>4th order curl operator using the central14 operator.  Factor: <EM>12h</EM>
<P>

<DT><CODE>curl2D(V)</CODE>
<DD>2nd order curl operator on a 2D vector field (e.g.
<CODE>Array&#60;TinyVector&#60;float,2&#62;,2&#62;</CODE>), using the central12 operator.  Factor:
<EM>2h</EM>
<P>

<DT><CODE>curl2D4(V)</CODE>
<DD>4th order curl operator on a 2D vector field, using the central12 operator.
Factor: <EM>12h</EM>
</DL>
<P>

Available in normalized forms <CODE>curln</CODE>, <CODE>curl4n</CODE>, <CODE>curl2Dn</CODE>,
<CODE>curl2D4n</CODE>.
</P><P>